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Fractions, Decimals and Percentages

Fractions, Decimals And Percentages

Here we will learn about fractions, decimals and percentages, including what they are, how to calculate with them and to solve problems involving them.

There are also fractions, decimals and percentages worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are fractions, decimals and percentages?

Fractions, decimals and percentages are different ways of representing a proportion of the same amount.

There is equivalence between fractions, decimals and percentages.

E.g.
43100=0.43=43%\frac{43}{100}=0.43=43\%

What are fractions, decimals and percentages?

What are fractions, decimals and percentages?

What are fractions?

Fractions are a way of writing equal parts of one whole.

They have a numerator (top number) and a denominator (bottom number). 
The denominator shows how many equal parts the whole has been divided into. 
The numerator shows how many of the equal parts we have.

E.g.
This shape has 9 9 equal parts and 4 4 of them are shaded.
This represents four ninths: 49\frac{4}{9}

Fractions, Decimals and Percentages image 1

What are decimals?

Decimals are a way of writing numbers that are not whole.

Decimal numbers can be recognised as they have a decimal point.
A decimal place is a position after the decimal point.

E.g.
0.37 0.37 has two decimal places.

There is a 3 3 in the tenths place and 7 7 in the hundredths place.

E.g.
This shows the fraction 710 \frac{7}{10}

Fractions, Decimals and Percentages image 2

710 \frac{7}{10} can also be written as 0.7 0.7

What are percentages?

Percentages are numbers which are expressed as parts of 100 100 .

Percent means “number of parts per hundred” and the symbol we use for this is the percent sign (%).

E.g. 43% 43\%

There are 100 100 equal parts and 43 43 of them are shaded.

Fractions, Decimals and Percentages image 3

How to use fractions, decimals and percentages

There are various ways of using fractions, decimals and percentages.

For examples, practice questions and worksheets on each one follow the links to the step by step guides below:

Comparing fractions, decimals and percentages

In order to compare fractions, decimals and percentages you need to be able to convert between them, including:

  • Converting fractions to decimals 
  • Converting decimals to fractions
  • Converting fractions to percentages
  • Converting percentages to fractions
  • Converting decimals to percentages
  • Converting percentages to decimals
  • Converting recurring decimals to fractions

Step-by-step guide: Converting fractions, decimals and percentages

Fractions, decimals and percentages worksheet

Fractions, decimals and percentages worksheet

Fractions, decimals and percentages worksheet

Get your free fractions, decimals and percentages worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Fractions, decimals and percentages worksheet

Fractions, decimals and percentages worksheet

Fractions, decimals and percentages worksheet

Get your free fractions, decimals and percentages worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Fractions, decimals and percentages examples

Fractions

1. Adding fractions

To add fractions they need to have a common denominator.

Show step

2. Subtracting fractions

To subtract fractions they need to have a common denominator.

Show step

3. Multiplying fractions

To multiply fractions we need to multiply the numerators together and multiply the denominators together.

Show step

4. Dividing fractions

To divide fractions we need to find the reciprocal of (flip) the second fraction, change the divide sign to a multiply and then multiply the fractions together.

Show step

5. Equivalent fractions

Equivalent fractions are fractions that have the same value

Show step

6. Improper fractions and mixed numbers

An improper fraction is a fraction where the numerator (top number) is larger than the denominator (bottom number).

A mixed number has a whole number part and a fractional part.

We can convert between improper fractions and mixed numbers:

Show step

7. Ordering fractions

To order fractions they need to have a common denominator.

Show step

8. Fractions of an amount

We can calculate a fraction of a given amount.

Show step

Decimals

1. Adding decimals

We can add decimals together:

Show step

2. Subtracting decimals

We can subtract decimals from each other:

Show step

3. Multiplying decimals

We can multiply decimals:

Show step

4. Dividing decimals

We can divide decimals by using equivalent fractions to ensure that the divisor (the denominator) is an integer:

Show step

Percentages

1. Percentage of an amount

We can find a percentage of an amount by breaking the percentage down:

Show step

2. Percentage multipliers

We can use percentage multipliers to find a percentage of an amount or to increase/decrease by a percentage:

Show step

3. Percentage increase

We can increase a value by a percentage:

Show step

4. Percentage decrease

We can decrease a value by a percentage:

Show step

5. Percentage change

We can calculate the percentage change between two values:

Show step

6. Reverse percentages

We can use reverse percentages to calculate the original number:

Show step

Comparing fractions, decimals and percentages

1. Fractions to decimals

Converting fractions to decimals:

Show step

2. Decimals to fractions

Converting decimals to fractions:

Show step

3. Fractions to percentages

Converting fractions to percentages:

Show step

4. Percentages to fractions

Converting percentages to fractions:

Show step

5. Decimal to percentage

Converting decimals to percentages:

Show step

6. Percentage to decimal

Converting percentages to decimals:

Show step

7. Recurring decimals to fractions

Converting recurring decimals to fractions:

Show step

Common misconceptions

  • Common denominators

To be able to add, subtract or compare fractions they must have a common denominator. To do this you need to find a common multiple for the denominators. The lowest common denominator is the easiest to use.

  • Fractions in their simplest form

Often fraction questions ask for the answer to be in its simplest form. This means you need to consider the numerator (the top number) and the denominator (the bottom number) and cancel by looking for common factors.

  • Percentages can be greater than 100100

Percentages can be more than 100 100 . This can happen for a percentage increase and for calculating percentage change.

  • The equivalence of one-third

Take care with one-third and its decimal and percentage equivalence.

\[ \begin{aligned} &\frac{1}{3}=0.3333333… = 33.33333…\%=33\frac{1}{3}\% \\\\ &33\% = 0.33 = \frac{33}{100} \end{aligned} \]

Practice fractions, decimals and percentage questions

1.  Calculate:

 

78    25 \frac{7}{8} \; − \; \frac{2}{5}

1940 \frac{19}{40}
GCSE Quiz True

53 \frac{5}{3}
GCSE Quiz False

11140 1\frac{11}{40}
GCSE Quiz False

2340 \frac{23}{40}
GCSE Quiz False
7825=35401640=351640=1940 \begin{aligned} &\frac{7}{8}-\frac{2}{5} \\\\ &=\frac{35}{40}-\frac{16}{40}\\\\ &=\frac{35-16}{40}\\\\ &=\frac{19}{40} \end{aligned}

2. Convert the following mixed number to an improper fraction:

 

235 2\frac{3}{5}

235 \frac{23}{5}
GCSE Quiz False

610 \frac{6}{10}
GCSE Quiz False

135 \frac{13}{5}
GCSE Quiz True

115 \frac{11}{5}
GCSE Quiz False
2×5=1010+3=13 \begin{aligned}2 \times 5 = 10\\\\ 10 + 3 = 13 \end{aligned}

3. Calculate:

 

2.8×1.3 2.8 \times 1.3

3.84 3.84
GCSE Quiz False

3.64 3.64
GCSE Quiz True

36.4 36.4
GCSE Quiz False

38.4 38.4
GCSE Quiz False

Fractions, Decimals and Percentages practice question 3

4. Calculate:

 

2.24÷0.4 2.24 \div 0.4

0.56 0.56
GCSE Quiz False

56 56
GCSE Quiz False

560 560
GCSE Quiz False

5.6 5.6
GCSE Quiz True

Fractions, Decimals and Percentages practice question 4

5.  Increase 45 45 by 12% 12\%

50.8 50.8
GCSE Quiz False

50.4 50.4
GCSE Quiz True

39.6 39.6
GCSE Quiz False

38.2 38.2
GCSE Quiz False
100%+12%=112%45×1.12=50.4 \begin{aligned} &100\% + 12\% = 112\% \\\\ &45\times 1.12=50.4 \end{aligned}

6. 65% 65\% of a number is 520 520 . What is the original number?

800 800
GCSE Quiz True

858 858
GCSE Quiz False

338 338
GCSE Quiz False

700 700
GCSE Quiz False

Fractions, Decimals and Percentages practice question 6

Fractions, decimals and percentages GCSE questions

1.  (a) Write 34 \frac{3}{4} as a decimal

 

(b) Write 0.7 0.7 as a fraction

 

(c) Write 0.6 0.6 as a percentage

 

(3 marks)

Show answer

(a) 34=0.75 \frac{3}{4}=0.75

(1)

 

(b) 0.7=710 0.7=\frac{7}{10}

(1)

 

(c) 0.6=610=60100=60% 0.6=\frac{6}{10}=\frac{60}{100}=60\%

(1)

2. Gordon buys a car.

The cost of the car is £13600 £13 600 plus VAT at 20% 20\%

 

Gordon pays a deposit of £4000 £4000

He pays the rest in 10 10 equal payments.

Work out the amount of each of the 10 10 payments.

 

(4 marks)

Show answer
20%= 2720 20\% = 2720

(for finding 20% 20\% of the price)

(1)

 

120%=16320 120\% = 16320

(for finding 120% 120\% of the price)

(1)

 

163204000=12320 16320-4000=12320

(for finding calculating the remainder to be paid)

(1)

 

12320÷10=1232 12320\div 10=1232

(for finding calculating the remainder to be paid)

(1)

3. Prove algebraically that the recurring decimal 0.43˙2˙ 0.4\dot{3}\dot{2} has the value of 214495 \frac{214}{495}

 

(3 marks)

Show answer
x=0.4323232 x=0.4323232…

(for the correct recurring decimal)

(1)

 

100x=43.232323.99x=42.8 \begin{aligned} &100x=43.232323…. \\\\ &99x=42.8 \end{aligned}

(for the second recurring decimal and the subtraction)

(1)

 

x=42.899=428990=214495 x=\frac{42.8}{99}=\frac{428}{990}=\frac{214}{495}

(for the correct fraction)

(1)

Learning checklist

You have now learned how to:

  • Order decimals and fractions
  • Understand and use place value for decimals
  • Use the 4 4 operations, including formal written methods, applied to decimals, proper and improper fractions, and mixed numbers
  • Define percentage as ‘number of parts per hundred’, interpret percentages and percentage changes as a fraction or a decimal, interpret these multiplicatively, express 1 1 quantity as a percentage of another, compare 2 2 quantities using percentages, and work with percentages greater than 100% 100\%
  • Convert fractions, decimals and percentages

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